In 1922 Achron emigrated to the United States, becoming a citizen in 1928. Achron's life changed on March 21, 1923 when he received an urgent telegram from Jascha Heifetz that read, "I am offering you a position as accompanist for tour beginning May 1, 1923 terminating May 1, 1924 including tour of the orient." For the next ten years Achron served as Heifetz's principal accompanist, recording an extensive discography and performing chamber and solo recitals in major music centers all over the world. On June 14, 1935 Achron married the mezzo-soprano Lea Karina. Karina, born in Helsinki in 1914, made her singing debut with the Helsingfors Symphony Orchestra in 1931 under the baton of Jean Sibelius. She was an accompolished pianist as well as singer who was able to perform vocal repertoire in nine different languages. In 1941 she made her New York debut at Town Hall with her husband accompanying. She went on to have a successful career in radio and later served as audio director for the popular television show Your Hit Parade. Mario Castelnuovo-Tedesco and Nicolas Slonimsky were among the composers who dedicated songs to her.

The mid 1930s witnessed Achron's return to solo performance as well as serious work as a composer. In 1937 he completed his First Concerto for Piano and Orchestra which he premiered on December 9, 1937 at Carnegie Hall with the New York Philharmonic, John Barbirolli conducting. He later performed the concerto with the London Symphony Orchestra and the Colonne Orchestra of Paris. Major publishers began to show interest in Achron's compositions with the publication of the First Piano Concerto and the first Sonnet for violin and piano in 1939 both by Carl Fischer, Inc. In 1937, Achron's song "Do You Know That The Moon Can Talk" was published by Chappell & Co., Ltd. under the pseudonym of Julian Dorr while the Suite Grotesque for piano was published in 1943 by G. Schirmer. Among those to whom Achron dedicated compositions were Heifetz, Yehudi Menuhin, Gregor Piatigorsky, and Mischa Elman. In 1946 Achron gave his final Carnegie Hall recital performing his own solo works as well as pieces by Mozart, Ravel, Liszt, and Chopin. He died suddenly on May 12, 1948 in New York City.